Properties

Label 4-2e18-1.1-c1e2-0-10
Degree $4$
Conductor $262144$
Sign $1$
Analytic cond. $16.7145$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 10·13-s + 16·17-s + 2·25-s + 6·29-s − 14·37-s + 14·49-s + 18·53-s − 22·61-s − 20·65-s − 9·81-s − 32·85-s − 16·97-s + 18·101-s − 26·109-s − 28·113-s − 10·125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.894·5-s + 2.77·13-s + 3.88·17-s + 2/5·25-s + 1.11·29-s − 2.30·37-s + 2·49-s + 2.47·53-s − 2.81·61-s − 2.48·65-s − 81-s − 3.47·85-s − 1.62·97-s + 1.79·101-s − 2.49·109-s − 2.63·113-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(262144\)    =    \(2^{18}\)
Sign: $1$
Analytic conductor: \(16.7145\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{512} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 262144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.044004972\)
\(L(\frac12)\) \(\approx\) \(2.044004972\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2^2$ \( 1 + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 + p^{2} T^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2^2$ \( 1 + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
59$C_2^2$ \( 1 + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2^2$ \( 1 + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.02386975674152690467768652483, −10.59295778641146198552345709771, −10.28848396900697739634015199402, −10.08246970809745146308831262477, −9.133028327215641590068972868978, −8.866597166253576373314699495777, −8.361559591341289211257769191423, −8.048848208304325473870910289010, −7.57355083188020345453515884079, −7.17765136673761200507377119936, −6.48900057728029578050946308496, −5.97668910654611376423357860269, −5.42686253922564345940117789280, −5.30213321231904925464493723223, −4.05222104021183707949451473760, −3.89294201188254270664626340478, −3.31615130913095394623605156135, −2.92793138425852203606671078724, −1.34762916831382432078830694610, −1.09270423426689080323935150257, 1.09270423426689080323935150257, 1.34762916831382432078830694610, 2.92793138425852203606671078724, 3.31615130913095394623605156135, 3.89294201188254270664626340478, 4.05222104021183707949451473760, 5.30213321231904925464493723223, 5.42686253922564345940117789280, 5.97668910654611376423357860269, 6.48900057728029578050946308496, 7.17765136673761200507377119936, 7.57355083188020345453515884079, 8.048848208304325473870910289010, 8.361559591341289211257769191423, 8.866597166253576373314699495777, 9.133028327215641590068972868978, 10.08246970809745146308831262477, 10.28848396900697739634015199402, 10.59295778641146198552345709771, 11.02386975674152690467768652483

Graph of the $Z$-function along the critical line